Controls how implicit arguments are handled in isDefEq during proof automation.
Original behavior (false): When simp, rw, or similar tactics apply a lemma, explicit
arguments are matched at the caller's transparency (typically .reducible). But implicit arguments
are "invisible" to the user — they don't choose them directly. If a lemma fails to apply because
an implicit argument doesn't match at .reducible, the user sees a confusing failure with no
obvious cause. To avoid this, we originally bumped transparency to .default for implicit arguments,
so that isDefEq would try harder and unfold semireducible definitions to make them match.
Why true is now the default: The transparency bump meant that every speculative isDefEq call
in proof automation could trigger expensive unfolding of semireducible definitions on implicit
arguments — and most of these calls fail. This eventually became a performance bottleneck in
Mathlib. With true, instance-implicit arguments ([..]) are checked at
TransparencyMode.instances (to resolve instance diamonds). Other implicit arguments are checked
at the caller's transparency unless backward.isDefEq.implicitBump is also true, in which case
all implicit arguments are bumped to .instances.
See isDefEqArgs for the implementation, backward.isDefEq.implicitBump for the implicit
argument bump, and TransparencyMode for the overall design.
Controls whether all implicit arguments (not just instance-implicit [..]) get their
transparency bumped to TransparencyMode.instances during isDefEq.
When true, all implicit arguments are checked at .instances, which unfolds
[implicit_reducible] definitions (instances, Nat.add, Array.size, etc.). This is the
intended behavior: users don't choose implicit arguments directly, so Lean should try harder
to make them match. The [implicit_reducible] attribute provides guardrails — only explicitly
marked definitions get unfolded, not arbitrary semireducible definitions.
When false (current default for staging), only instance-implicit arguments ([..]) are bumped
to .instances; other implicit arguments stay at the caller's transparency.
This option only has an effect when backward.isDefEq.respectTransparency is true.
Return true if e is of the form fun (x_1 ... x_n) => ?m y_1 ... y_k), and ?m is unassigned.
Remark: n, k may be 0.
This function is used to filter unification problems in
isDefEqArgs/isDefEqEtaStruct where we can assign proofs.
If one side is of the form described above, then we can likely assign ?m.
But if it's not, we would most likely apply proof irrelevance, which is
usually very expensive since it needs to unify the types as well.
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- Lean.Meta.isAbstractedUnassignedMVar (Lean.Expr.lam binderName binderType b binderInfo) = Lean.Meta.isAbstractedUnassignedMVar b
- Lean.Meta.isAbstractedUnassignedMVar (a.app arg) = Lean.Meta.isAbstractedUnassignedMVar a
- Lean.Meta.isAbstractedUnassignedMVar x✝ = pure false
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Support for Lean.reduceBool and Lean.reduceNat
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Support for reducing Nat basic operations.
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Support for constraints of the form ("..." =?= String.ofList cs)
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Return true if e is of the form fun (x_1 ... x_n) => ?m x_1 ... x_n), and ?m is unassigned.
Remark: n may be 0.
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Result type for isDefEqArgsFirstPass.
- failed : DefEqArgsFirstPassResult
Failed to establish that explicit arguments are def-eq. Remark: higher output parameters, and parameters that depend on them are postponed.
- ok
(postponedImplicit postponedHO : Array Nat)
: DefEqArgsFirstPassResult
Succeeded. The array
postponedImplicitcontains the position of the implicit arguments for which def-eq has been postponed.postponedHOcontains the higher order output parameters, and parameters that depend on them. They should be processed after the implicit ones.postponedHOis used to handle applications involving functions that contain higher order output parameters. Example:getElem : {cont : Type u_1} → {idx : Type u_2} → {elem : Type u_3} → {dom : cont → idx → Prop} → [self : GetElem cont idx elem dom] → (xs : cont) → (i : idx) → (h : dom xs i) → elemThe arguments
domandhmust be processed after all implicit arguments otherwise higher-order unification problems are generated. See issue #1299, when trying to solvegetElem ?a ?i ?h =?= getElem a i (Fin.val_lt_of_le i ...)we have to solve the constraint
?dom a i.val =?= LT.lt i.val (Array.size a)by solving after the instance has been synthesized, we reduce this constraint to a simple check.
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Ensure MetaM configuration is strong enough for checking definitional equality of
instances. For example, we must be able to unfold instances, beta := true, proj := .yesWithDelta
are essential.
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Check whether the types of the free variables at fvars are
definitionally equal to the types at ds₂.
Pre: fvars.size == ds₂.size
This method also updates the set of local instances, and invokes
the continuation k with the updated set.
We can't use withNewLocalInstances because the isDefEq fvarType d₂
may use local instances.
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- Lean.Meta.isDefEqBindingDomain fvars ds₂ k = Lean.Meta.isDefEqBindingDomain.loop✝ fvars ds₂ k 0
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Each metavariable is declared in a particular local context.
We use the notation C |- ?m : t to denote a metavariable ?m that
was declared at the local context C with type t (see MetavarDecl).
We also use ?m@C as a shorthand for C |- ?m : t where t is the type of ?m.
The following method process the unification constraint
?m@C a₁ ... aₙ =?= t
We say the unification constraint is a pattern IFF
1) `a₁ ... aₙ` are pairwise distinct free variables that are *not* let-variables.*not* let-variables. let-variables.
2) `a₁ ... aₙ` are not in `C`
3) `t` only contains free variables in `C` and/or `{a₁, ..., aₙ}`
4) For every metavariable `?m'@C'` occurring in `t`, `C'` is a subprefix of `C`
5) `?m` does not occur in `t`
Claim: we don't have to check free variable declarations. That is,
if t contains a reference to x : A := v, we don't need to check v.
Reason: The reference to x is a free variable, and it must be in C (by 1 and 3).
If x is in C, then any metavariable occurring in v must have been defined in a strict subprefix of C.
So, condition 4 and 5 are satisfied.
If the conditions above have been satisfied, then the solution for the unification constrain is
?m := fun a₁ ... aₙ => t
Now, we consider some workarounds/approximations.
A1) Suppose t contains a reference to x : A := v and x is not in C (failed condition 3)
(precise) solution: unfold x in t.
A2) Suppose some aᵢ is in C (failed condition 2)
(approximated) solution (when config.quasiPatternApprox is set to true) :
ignore condition and also use
?m := fun a₁ ... aₙ => t
Here is an example where this approximation fails:
Given C containing a : nat, consider the following two constraints
?m@C a =?= a
?m@C b =?= a
If we use the approximation in the first constraint, we get ?m := fun x => x when we apply this solution to the second one we get a failure.
IMPORTANT: When applying this approximation we need to make sure the
abstracted term fun a₁ ... aₙ => t is type correct. The check
can only be skipped in the pattern case described above. Consider
the following example. Given the local context
(α : Type) (a : α)
we try to solve
?m α =?= @id α a
If we use the approximation above we obtain:
?m := (fun α' => @id α' a)
which is a type incorrect term. a has type α but it is expected to have
type α'.
The problem occurs because the right hand side contains a free variable
a that depends on the free variable α being abstracted. Note that
this dependency cannot occur in patterns.
We can address this by type checking the term after abstraction. This is not a significant performance bottleneck because this case doesn't happen very often in practice (262 times when compiling stdlib on Jan 2018). The second example is trickier, but it also occurs less frequently (8 times when compiling stdlib on Jan 2018, and all occurrences were at Init/Control when we define monads and auxiliary combinators for them). We considered three options for the addressing the issue on the second example:
A3) a₁ ... aₙ are not pairwise distinct (failed condition 1).
In Lean3, we would try to approximate this case using an approach similar to A2.
However, this approximation complicates the code, and is never used in the
Lean3 stdlib and mathlib.
A4) t contains a metavariable ?m'@C' where C' is not a subprefix of C.
If ?m' is assigned, we substitute.
If not, we create an auxiliary metavariable with a smaller scope.
Actually, we let elimMVarDeps at MetavarContext.lean to perform this step.
A5) If some aᵢ is not a free variable,
then we use first-order unification (if config.foApprox is set to true)
?m a_1 ... a_i a_{i+1} ... a_{i+k} =?= f b_1 ... b_k
reduces to
?M a_1 ... a_i =?= f
a_{i+1} =?= b_1
...
a_{i+k} =?= b_k
A6) If (m =?= v) is of the form
?m a_1 ... a_n =?= ?m b_1 ... b_k
then we use first-order unification (if `config.foApprox` is set to true)
A7) When foApprox, we may use another approximation (constApprox) for solving constraints of the form
?m s₁ ... sₙ =?= t
```
where s₁ ... sₙ are arbitrary terms. We solve them by assigning the constant function to ?m.
?m := fun _ ... _ => t
In general, this approximation may produce bad solutions, and may prevent coercions from being tried.
For example, consider the term `pure (x > 0)` with inferred type `?m Prop` and expected type `IO Bool`.
In this situation, the
elaborator generates the unification constraint
```
?m Prop =?= IO Bool
```
It is not a higher-order pattern, nor first-order approximation is applicable. However, constant approximation
produces the bogus solution `?m := fun _ => IO Bool`, and prevents the system from using the coercion from
the decidable proposition `x > 0` to `Bool`.
On the other hand, the constant approximation is desirable for elaborating the term
```
let f (x : _) := pure "hello"; f ()
```
with expected type `IO String`.
In this example, the following unification constraint is generated.
```
?m () String =?= IO String
```
It is not a higher-order pattern, first-order approximation reduces it to
```
?m () =?= IO
```
which fails to be solved. However, constant approximation solves it by assigning
```
?m := fun _ => IO
```
Note that `f`s type is `(x : ?α) -> ?m x String`. The metavariable `?m` may depend on `x`.
If `constApprox` is set to true, we use constant approximation. Otherwise, we use a heuristic to decide
whether we should apply it or not. The heuristic is based on observing where the constraints above come from.
In the first example, the constraint `?m Prop =?= IO Bool` come from polymorphic method where `?m` is expected to
be a **function** of type `Type -> Type`. In the second example, the first argument of `?m` is used to model
a **potential** dependency on `x`. By using constant approximation here, we are just saying the type of `f`
does **not** depend on `x`. We claim this is a reasonable approximation in practice. Moreover, it is expected
by any functional programmer used to non-dependently type languages (e.g., Haskell).
We distinguish the two cases above by using the field `numScopeArgs` at `MetavarDecl`. This field tracks
how many metavariable arguments are representing dependencies.
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- Lean.Meta.mkAuxMVar lctx localInsts type numScopeArgs = Lean.Meta.mkFreshExprMVarAt lctx localInsts type Lean.MetavarKind.natural Lean.Name.anonymous numScopeArgs
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- mvarId : MVarId
- mvarDecl : MetavarDecl
- hasCtxLocals : Bool
- rhs : Expr
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Auxiliary function used to "fix" subterms of the form ?m x_1 ... x_n where x_is are free variables,
and one of them is out-of-scope.
See Expr.app case at check.
If ctxApprox is true, then we solve this case by creating a fresh metavariable ?n with the correct scope,
an assigning ?m := fun _ ... _ => ?n
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- Lean.Meta.CheckAssignmentQuick.check hasCtxLocals mctx lctx mvarDecl mvarId fvars e = Lean.Meta.CheckAssignmentQuick.check.unsafe_impl_3✝ hasCtxLocals mctx lctx mvarDecl mvarId fvars e
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Auxiliary function for handling constraints of the form ?m a₁ ... aₙ =?= v.
It will check whether we can perform the assignment
?m := fun fvars => v
The result is none if the assignment can't be performed.
The result is some newV where newV is a possibly updated v. This method may need
to unfold let-declarations.
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- Lean.Meta.whenUndefDo x k = do let __do_lift ← x match __do_lift with | Lean.LBool.true => pure true | Lean.LBool.false => pure false | Lean.LBool.undef => k
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Result type for isDefEqDelta
- eq : DeltaStepResult
- unknown : DeltaStepResult
- cont (t s : Expr) : DeltaStepResult
- diff (t s : Expr) : DeltaStepResult
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- transient : DefEqCacheKind
- permanent : DefEqCacheKind
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Structure for storing defeq cache key information.
- kind : DefEqCacheKind
- key : DefEqCacheKey